Integrand size = 29, antiderivative size = 47 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\frac {1}{2} a (A+2 B) x+\frac {a (A+B) \sin (c+d x)}{d}+\frac {a A \cos (c+d x) \sin (c+d x)}{2 d} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {4081, 3872, 2717, 8} \[ \int \cos ^2(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\frac {a (A+B) \sin (c+d x)}{d}+\frac {1}{2} a x (A+2 B)+\frac {a A \sin (c+d x) \cos (c+d x)}{2 d} \]
[In]
[Out]
Rule 8
Rule 2717
Rule 3872
Rule 4081
Rubi steps \begin{align*} \text {integral}& = \frac {a A \cos (c+d x) \sin (c+d x)}{2 d}-\frac {1}{2} \int \cos (c+d x) (-2 a (A+B)-a (A+2 B) \sec (c+d x)) \, dx \\ & = \frac {a A \cos (c+d x) \sin (c+d x)}{2 d}+(a (A+B)) \int \cos (c+d x) \, dx+\frac {1}{2} (a (A+2 B)) \int 1 \, dx \\ & = \frac {1}{2} a (A+2 B) x+\frac {a (A+B) \sin (c+d x)}{d}+\frac {a A \cos (c+d x) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\frac {a (2 A c+2 A d x+4 B d x+4 (A+B) \sin (c+d x)+A \sin (2 (c+d x)))}{4 d} \]
[In]
[Out]
Time = 0.71 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.89
method | result | size |
parallelrisch | \(\frac {\left (\frac {A \sin \left (2 d x +2 c \right )}{2}+\left (2 A +2 B \right ) \sin \left (d x +c \right )+\left (A +2 B \right ) x d \right ) a}{2 d}\) | \(42\) |
risch | \(\frac {a A x}{2}+a x B +\frac {a A \sin \left (d x +c \right )}{d}+\frac {\sin \left (d x +c \right ) B a}{d}+\frac {a A \sin \left (2 d x +2 c \right )}{4 d}\) | \(51\) |
derivativedivides | \(\frac {a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )+B a \sin \left (d x +c \right )+B a \left (d x +c \right )}{d}\) | \(57\) |
default | \(\frac {a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )+B a \sin \left (d x +c \right )+B a \left (d x +c \right )}{d}\) | \(57\) |
norman | \(\frac {\frac {a \left (A +2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {a \left (A +2 B \right ) x}{2}+\frac {2 a A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-\frac {a \left (A +2 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+\frac {a \left (A +2 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}+\frac {a \left (A +2 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}-\frac {a \left (3 A +2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\) | \(163\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\frac {{\left (A + 2 \, B\right )} a d x + {\left (A a \cos \left (d x + c\right ) + 2 \, {\left (A + B\right )} a\right )} \sin \left (d x + c\right )}{2 \, d} \]
[In]
[Out]
\[ \int \cos ^2(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx=a \left (\int A \cos ^{2}{\left (c + d x \right )}\, dx + \int A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.17 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a + 4 \, {\left (d x + c\right )} B a + 4 \, A a \sin \left (d x + c\right ) + 4 \, B a \sin \left (d x + c\right )}{4 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (43) = 86\).
Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.98 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\frac {{\left (A a + 2 \, B a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
[In]
[Out]
Time = 13.52 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.06 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\frac {A\,a\,x}{2}+B\,a\,x+\frac {A\,a\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a\,\sin \left (c+d\,x\right )}{d}+\frac {A\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
[In]
[Out]